Te Puna Ako Learning Centre
Rearranging (Transposing) Formulae or Changing the Subject of a Formula
The subject of a formula is the “letter” on its own on one side of the formula. While a
formula is usually given with the same subject in fact the formula can be used to find any
of the “letters” in the formula. To do this we change the subject of the formula.
For example the formula to find the area of a rectangle is A = lw where A is the area of the
rectangle, l is the length of the rectangle and w is the width of the rectangle. When the
formula is in this form we can find the area of the rectangle if we know the length and the
width. However we can also use this same formula to find the length of the rectangle if we
know the area and the width or the width of the rectangle if we know the area and the
length. To do this we change the subject of the formula.
Note: Where there is no operation sign in the formula the operation is multiplication
l + w, l – w,
l
w
and lw (means l x w)
[ l(w + 3) means l x ( w + 3) ]
How to Change the Subject of a Formula
(a)
Leave the new subject where it is (unless it is negative – we will discuss this later)
(b)
Undo what is done to the new subject by moving letters and numbers across the
equal sign remembering to do the opposite operation on the other side.
Opposite operations:
addition and subtraction
multiplication and division
squaring and square rooting etc.
(c)
Generally operations are “undone” in the opposite order to that in which they are
“done”. (The opposite of BEDMAS). That is addition and subtraction are undone
first, followed by multiplication and division and finally powers and roots.
Unitec: Document1
For example:
1. Make u the subject of: v = u + at
2. Make t the subject of:
v – at = u
v = u + at
v – u = at
v u
t
a
3. Make a the subject of: v = at 2 - u
4. Make t the subject of:
v = at 2 - u
v + u = at 2
v + u = at 2
v u
a
t2
v u
 t2
a
v u
t
a
Exceptions to the Order Rule
(a)
If there is an operation that is done to the whole side the new subject is on it must
be undone first.
For example:
5. Make u the subject of: v =
u  at
6. Make t the subject of:
v = (u  at) 2
v 2 = u + at
v  u  at
v 2 - at = u
v - u = at
v u
t
a
7. Make a the subject of: v =
ua
t
vt = u + a
vt – u = a
Unitec: Document1
(b)
If the new subject is a denominator or part of a denominator it must be moved off
the bottom by multiplying on the other side before you attempt to make it the
subject.
For example:
8. Make t the subject of:
v=
ua
t
vt = u + a
t=
ua
v
9. Make t the subject of:
v=
v(t  1)2  u  a
(t  1) 2 
ua
v
t+1=
ua
v
t=
(c)
ua
(t  1) 2
ua
1
v
If the subject is negative move it to the other side of the equation first to make it
positive
For example:
10. Make u the subject:
v = at – u
v + u = at
u = at – v
Unitec: Document1
Rearranging Formulae Exercise Sheet
Make the letter in brackets the subject of the following equations
Exercise A
1. G + 2R = 630mm (G)
4. A =
1
bh (h)
2
2. G + 2R = 630mm (R)
3. A = lw (l)
5. m = b²
6. m = 4 - b
Exercise B
1. v = u + at (u)
4. I =
PRT
100
(R)
2. v = u + at (t)
3. A = r 2 (r)
5. s = 2r(r + h) (h)
6. v² = u² + 2as (u)
Answers:
Exercise A
1 G = 630mm – 2R
2. R =
630mm  G
2
5. b =
m
A
3. l 
4. h 
2A
w
b
6. B = 4 - m
Exercise B
1. u = v – at
5. h 
s  2r
2.
t=
v u
3. r =
a
2
2r
Unitec: Document1
or h 
s
2r
r
6. u =
A

2
v  2as
4. R =
100 I
PT
(A 
1
2
=Ax
2
1
)